On the Uniqueness of L²-solutions in half-space of certain differential equations

Square integrable solutions to the equation {-∂²/∂y² + P(D_x)+b(y)-λ}u(x, y) = f(x, y) are considered in the half-space y>0, x ∈ℝⁿ, where P(D_x) is a constant coefficient operator. Under suitable conditions on lim_y→0u(x, y), b(y), f(x, y) and λ, it is shown that supp u = supp f. This generalizes a result due to Walter Littman.